1. Worst-case Fair Weighted Fair Queueing (WF²Q) by Jon C.R. Bennett & Hui Zhang Presented by Vitali Greenberg

2. 1.Provides an end-to-end bounded-delay service to a session whose traffic is constrained by a leaky bucket. 2.Ensures fair allocation of bandwidth among all backlogged sessions regardless of whether their traffic is constrained or not. 3.Unrealizable…=> approximation algorithms as Weighted Fair Queueing (WFQ) known also as Packet Generalized Processor Sharing (PGPS). The Generalized Processor Sharing (GPS) discipline.

3. Definition 1 Two queuing systems with different service disciplines are called corresponding systems of each other if they have the same speed, same set of sessions, same arrival pattern, and if applicable, same service share for each session.

4. GPS & WFQ GPS is an idealized server that does not transmit packets as entities. It assumes that the server can serve all backlogged sessions simultaneously and that the traffic is infinitely divisible. In WFQ, when a server ready to transmit the next packet at time , it picks the first packet that would complete service in the corresponding GPS system if no additional packets arrived after time .

5. A separate FIFO queue for each session sharing the same link. During any time interval, when there are N non- empty queues, GPS allows different session to have different service shares and serves the non- empty queues in proportion to their session’s service share simultaneously. A GPS server serving N session is characterized by N positive real numbers:  1,  2, …,  N. The server operates at a fixed rate r and is work- conserving. About GPS in general…

6. Example GPS  WFQ  Arriving function 

7. Example (Cont.) Such oscillation is undesirable for feedback-based congestion control algorithms in data communication networks. A data source has to balance between two considerations: on the one hand, it wants to send data to the network as fast as possible, on the other hand, it doesn’t want to send the data so fast that causes network congestion. To achieve the best performance, the source needs to detect the amount of bandwidth available to itself and match its sending rate to the available bandwidth.

8. Definition 2 A service discipline s is called worst-case fair for session i if for any time , the delay of a packet arriving at  is bounded above by: Where r i is the throughput guaranteed to session i, Q i,s (  ) is the queue size of session i at time , and C i,s is a constant independent of the queues of the other sessions sharing the multiplexer. A service discipline is called worst-case fair if it is worst-case fair for all sessions. C i,s is called Worst-case Fair Index for session i at server s.

9. Since C i,s is measured in absolute time, it is not suitable for comparing C i,s `s of sessions with different r i `s. Let’s define Normalized Worst-case Fair Index as: And for server that is worst-case fair, we define its Normalized Worst-case Fair Index as: Notice that GPS is worst-case fair with c GPS = 0 and c WFQ may increase linearly as a function of number of sessions.

10. Worst-case Fair Weighted Fair Queuing (WF²Q) WF²Q as packet approximation policy of GPS. In a WFQ system, when the server chooses the next packet to transmission at time , it selects the first packet that would complete service in the corresponding GPS system. In a WF²Q system the server chooses the next packet to transmission among those which already started and selects the packet that would complete service first in the corresponding GPS system.

11. and the picture would look so…

12. Theorem 1 Given a WF²Q system and a corresponding GPS system, the following properties hold for any i, k,  :

13. Theorem 1 - Proof

14. In this paper, we consider two rate-controlled service disciplines: R­WFQ and R­GPS, which have the same regulators but different schedulers. The schedulers for R­WFQ and R­GPS are WFQ and GPS respectively. Therefore, R­WFQ is a packet algorithm and R­GPS is a fluid algorithm. The eligibility time for the k-th packet on session i is defined to be: where is the time the packet starts service in the corresponding GPS system.

15. Theorem 1 - Proof Notice that there are two GPS servers under consideration, the corresponding GPS server that is standalone, and the GPS server that is embedded within the R­GPS server. To distinguish between them, we refer to the embedded one as GPS*. Likewise, we refer to the embedded WFQ server in R­WFQ as WFQ*.

16. Theorem 1 - Proof Lemma 1: An R­GPS system is equivalent to its corresponding GPS system, i.e., for any arrival sequence, the instantaneous service rates for each connection at any given time are exactly the same with either service discipline, and holds. Lemma 2: An R­WFQ system is equivalent to the corresponding WF²Q system, i.e., for any arrival sequence, packets are serviced in exactly the same order with either service discipline and holds.

17. Theorem 1 - Proof

18. Since WF²Q is equivalent to the corresponding R-WFQ and GPS system is equivalent to the corresponding R-GPS one we’ll show that

19. Theorem 1 - Proof Since the input traffic, the regulators, and service shares for all sessions are identical for the two corresponding R-WFQ and R-GPS systems, the input traffic and the per session service shares for the two embedded WFQ* and GPS* systems are also identical. Therefore the embedded WFQ* and GPS* are also corresponding systems. That proves the first two inequalities.

20. Theorem 1 - Proof Since a packet will not start service in a WF²Q system until it starts service in the corresponding GPS system, the following must hold (1) Without losing generality, let Since the maximum number of bits that can be served during the interval by WF²Q is limited by both the link speed and the packet size, we have: (2)

21. Theorem 1 - Proof Also, since GPS guarantees a service rate to a backlogged session, we have: (3) Combining (2) and (3), we have: the right side is maximized when or when

22. Theorem 1 - Proof So maximizing the right side we have: Combining the last inequality and (1) we receive: Which proves the third inequality and the theorem.

23. Since the backlog function is the difference between the cumulative arrival function and cumulative service function, the fact that the service functions of WF²Q and GPS are close, implies that their backlog functions are also very close.

24. Corollary 1 For two corresponding WF²Q and GPS systems the following holds :

25. Theorem 2 WF²Q is worst-case fair for session i with Worst-case Fair Index of: where: - is the session i maximum packet size. - is the maximum packet size among all sessions. - is the session i guaranteed transmission rate. - is the link speed.

26. Corollary 2 In a network with all packets having the same size L, such as ATM network, WF²Q is worst-case fair for session i with Worst-case Fair Index of: WF²Q is worst-case fair with the Normalized Worst- case Fair Index of: Note that WF²Q is a work-conserving discipline.

27. Related work Self-Clocked Fair Queuing (SCFQ) – simpler, larger delay bound. Virtual Clock algorithm – identical delay bound, Worst-case Fair Index can be arbitrary large.

28. Summary Contrary to popular belief, there can be a large discrepancy between the service provided by a packet WFQ system and the fluid GPS one. A new packet approximation algorithm of GPS called Worst-case Fair Weighted Fair Queuing (WF²Q) that provides almost identical service to GPS differing by no more than one maximum size packet.